Physica D 36 (1989) 209-221
North-Holland, Amsterdam
A CELLULAR AUTOMATON DESCRIBING THE FORMATION OF SPATIALLY ORDERED
STRUCTURES IN CHEMICAL SYSTEMS
M. GERHARDT and H. SCHUSTER
Universit~t Bielefel& Fakultiit f~r Mathematik, D-4800 Bielefeld !, Postfach 8640, Fed. Rep. Germany
Received 10 February 1988
Revised manuscript received 22 December 1988
Communicated by F.H. Busse
A cellular automaton describing a certain heterogeneous catalytic reaction is introduced as a theoretical approach towards
the investigation of pattern formation in chemical systems. The numerically obtained results demonstrate that the time
evolution of this cellular automaton leads to a self-sustained organization of circular and spiral wave-like structures, even
starting from random initial states.
I. Introduction
r00
In the fellowing we want to introduce a mathematical model which we developed for a certain
heterogeneous catalytic reaction, namely the oxidation of carbon monoxide catalyzed by palladium crystallites which are incorporated into a
zeolite matrix. This reaction was empirically investigated by Jaeger et al. [1-31. Complex oscil!afion
patterns could be observed in this chemical system, their form depending on the experimental
conditions, see fig. 1.
Mathematical models of heterogeneous catalytic
reactions are usually based on the assumption of
the uniformity of the catalyst and of the rates of
processes which occur thereon. But different experimental results obtained in the last years
demonstrated that spatial variations on the catalyst have to be considered [4-9].
Therefore, we were mainly interested in the
question whether the complex oscillation patterns
of the aforementioned CO oxidation might be
correlated to the formation of certain spatially
ordered structures on the catalyst, which are
caused by local interactions between different
I
t
7C t n ~,'~
t
Fig. 1. Experimentally observed oscillation patterns of the
CO_,-production (upper trace) and the temperature during the
oxidation of CO on palladium-loaded zeolites [1-3].
"catalytic units." For the mathematical description of this chemical system we chose the approach of cellular automata which, above all,
reflect the idea of local interactions between large
numbers of single sub-systems.
Ever since John von Neumann introduced the
idea of a cellular automaton [I0], the theory and
applications of cellular automata have become an
important tool for a theoretical approach towards
complex systems consisting of a large number of
interacting particles [11-13]. In spite of the si, 1plicity of their construction cellular automata ca
exhibit complex dynamical behaviour. Even if the
0167-2789/89/$03.50 © Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
210
M. Gerhardt and H. Schuster / A cellular automaton
analysis of their behaviour is hardly possible by
traditional mathematical methods, cellular automata - as complete discrete dynamical systems are most directly suitable for implementation on
conventional digital computers to allow their investigation by numerical calculations.
In using cellular automata as theoretical models
,or natural systems one must be aware of their
discrete nature. Such a discrete approach seems to
us to be appropiate in describing the specific situation of the aforementioned CO oxidation. This is,
for example, justified by the fact that the catalyst
consists of a large number of palladium crystallites and can, therefore, be considered as a discretely distributed set of such catalytic units
corresponding directly to the discrete set of cells
of a cellular automaton.
During the n,.,merical investigation of the cellular automaton which we designed, it turned out
that this automaton not only describes the typical
behaviour of the CO oxidation, but that it leads to
a self-sustained organization of fascinating spatial
patterns, such as circular and spiral waves (cf.
plate I). rhese are, to our surprise, very similar
to those observed in excitable reed, a, e.g. the
Belousov-Zhabotinskii reaction [14-19 i
In section 2 of this paper the basic chemica!
ideas according to the reaction mechanism of the
CO oxidation are briefly summarized. The :ellular
automaton which we defined and studied and
which we like to call the "hodge podge mac/fine"
(cf. [201) is presented in the ensuing section. In
section 4 the dynamical behaviour of this cellmar
automaton is described on the basis of humeri.
cally obtained time evolutions which start from
randomly determined initial states. In the last
section the significance of these results is discussed
with respect to the chemical background.
2. CO oxidation
Oscillations in the rate of CO 2 production over
different metal catalysts, like piatinum or palladium, have often been reported m the last years
[5-8, 21-27]. A variety of explanations have been
proposed in the literature to explain the oscillatory behaviour of th~s reaction.
It is generally accepted that CO oxidation proceeds by means of the Langmuir-Hinshelwood
(LH) mechanism, in which both CO and O must
be chemisorbed to the catalyst surface before a
CO 2 molecule can be produced. Simple models
describing the reaction kinetics have shown that
muldple steady states arise from the LH mechanism [25].
Oscillations in the rate of CO 2 production are
mostly assumed to be correlated to transitions
betweer~ two different reaction branches, whereas
the upper reaction branch is characterized by a
small CO coverage on the catalyst surface while
the lower reaction branch is related to a high CO
coverage. An explanation for the oscillatory beha,viour of t~e CO oxidation has te describe the
,driving mechanism which is responsible for the
transitions between these two (stable) reaction
br,uaches. Some authors suggest that these oscillati.ons are correlated to different surface structures
of the acsorbed mol~ules, as it arises from COclustenag [22, 23] or an adsorbate induced structural tra~:~formation of the catalyst surface atoms
between dfferent phases [6, 7]. Computer simulations based on models in the form of cellular
automata ,m wifich each cell describes a single
adsorption site on the catalyst surface could support the relevance of these concep,'s for the discussion of the CO oxidation [7, 28, 29]. Besides these
concepts other authors regard an oxidation process of the metal surface as the essential driving
mechanism for the observed oscillations in different catalytic systems [25, 26, 30].
For the specific CO oxidation investigated by
Jaeger et al. [1-3], it was also postulated that the
production of CO 2 is only possible by chemisorption of CO and O on the catalyst surface as
described by the LH mechanism. Following Turner
et a I. [25, 26] a cyclic oxidation-reduction mechal:iSql was , ugge~ted as an explanation of the oscillato2,' behaviour of this catalytic system. This
meclanism was supported by the experimental
M. Gerhardt and H. Schuster / A cellular automaton
observation of palladiumoxide reflexes obtained
by X-ray studies of catalyst samples after the
reaction [3].
Since the X-ray diffraction patterns showed reflexes of metal and metaloxide simultaneously, it
is obvious to assume that the catalyst consisting of
a large number of palladium crystallites does not
act in complete synchrony. By correlating the different size of the peaks of decrease in the CO
conversion with a different number of palladium
crystallites which are simultaneously inhibited for
the CO 2 production, the observation of the characteristic oscillation patterns, as depicted in fig. 1,
suggests that the behaviotu of this catalytic system
is determined by the formation of certain spatially
ordered structures on the catalyst during its time
evolution. Local interactions between different
catalytic units in the form of heat transfer and CO
diffusion are assumed to be the cause for the
formation of such spatial patterns on the catalyst.
It was our main interest to develop a mathematical model for the theoretical discussion of such a
pattern formation induced by local interactions
between single subsystems. The rules of the cellular au!omaton introduced in the following section
are derived from the postulate,,a reaction scheme
of the CO oxidation. The evolution o,~ a single
subsystem (cell) is based on the general idea of a
transition cycle between two different reaction
branches and does not explicitly consider the adsorption processes of the involved molecules on
the catalyst. This rather abstract description of the
reaction mechanism of a sin~e catalytic unit enables the simulation of an interaction between a
large number of such units.
3. The cellular automaton
The cellular automaton should describe first of
all the afo:-ementio.,'.e~ CO oxidation. I:oi rids
reason its rules are derived from the postulated
reaction scheme of this catalytic system.
The catalyst used in this experiment consisted
of a large number of palladium crystallites. Con-
211
sidering the reaction mechanism of a single palladium crystallite, a reversible phase transition between an active palladium phase and an inactive
palladiumoxide phase was postulated. The essential idea of the following cellular automaton is
based on the chemical assumption that a palladium crystallite in the palladium phase can be
excited by neighbouring particles by way of local
interactions to form the palladiumcxide phase.
This excitation is referred to as an infection process.
The development of an excited catalytic unit
from the palladium phase to the palladiumoxide
phase may then be considered as a storing process,
whereby the interior of the o'ystallites is gradually
filled up with oxygen until a certain threshold
concentration is reached, which characterizes the
inactive palladiumoxide phase.
According to these chemical ideas the cellular
automaton is defined as follows: A cell of the
cellular automaton is regarded as a single catalytic
unit, e.g. a single palladium crystallite or a strongly
coupled set of palladium crystaUites which act in
complete synchrony.
The set X of cells is given by a two-din'ensi,~llal
rectilinear n x m lattice:
X= {(i,j)ll<_i<n,1 <j<_m},
n,mG .
On this lattice two well-known forms of neighbourhood are considered: the yon Neumann
neighbourhood and the Moore neighbourhood in
which each cell has four and eight neigh0ours,
resp. (cf. fig. 2). The set f~ of possible states of a
cell is chosen to be a finite subset of the natural
numbers IN:
~ = {0,1,2 . . . . . V}
with VE N.
(2)
The states of a cell are interpreted as follows: The
state characterized by the value 0 represents the
pure palladium phase of a catalytic unit and is
212
M. Gerhardt and H. Schuster / A cellular automaton
step the number of its ill and infected neighbours
by Ku(t) and l,j(t), respectively. Ku(t ) and lu(t)
are then given by
K , j ( ' ) = # {(r, s) ~ X I (r, s) is neighbouring
a)
Fig. 2. Two forms of neighbourhood in two-dimensional cellular automata: (a) "yon Neumann'" and (b) "Moore" neighbourhocd. The white squares symbolize the cells neighbouring
the black center cell. The center cell is always considered as a
neighbour of itself.
called "healthy." The state given by the value V
represents the palladiumoxide phase and is called
"ill." All other values between 1 and V - 1 describe the state of the catalytic unit developing
into the palladiumoxide phase. Such a state is
called "im'ected."
The state evolution of a cell x,j is recursively
defined in terms of these three different infection
states:
3.1. The evolution of a healthy cell
Based on the chemical concepts, it is assumed
that a catalytic unit in the palladium phase can be
excited to form the palladiumoxide phase by two
mechanisms. First, it can be excited by heat transfer from neighbouring particles which are already
developing into the palladiumoxide phase and are
therefore in a more reactive state having higher
temperatures. Secondly, it can be excited by
neighbouring particles which have already reached
the palladiumoxide phase. This is possible by way
of CO diffusion, i.e. by the supply of additional
CO which can no longer be converted on the
inactive palladiumoxide phase. To reflect these
chemical concepts of interaction, a healthy cell
can become infected, if a certain number of infected or ill cells are present in its neighbourhood.
So, if the cell x;. is healthy at time t, we count
for the computation of its state at the next time
t o ( i , j ) and x , , ( t ) = V},
(3)
I tj ( t ) = # { ( r, s ) E X I ( r, s) is neighbouring
to (i, j ) a n d 0 < x , , ( t ) < V}.
(4)
If these numbers are greater than certain values
k~ ~ I%1 and k 2 ~ ~1, respectively, the healthy cell
gets infected. The state evolution of a healthy cell
is then formalized by
%(t + 1)
[
K~j(t)
=
(5)
In this and the following expressions, [ ] stands
for the Gaussian bracket which designates a
rouading-down process applied to the fractions
they contain. If, for example, K u ( t ) / k I happens
to be equal to 2.5, the Gaussian bracket reduces
that number to 2.
3.2. The evolution of an infected cell
An infected cell should gradually reach the ill
state. In the cellular automaton, this is expressed
in such a way that the degree of infection of a cell
linearly increases in time, i.e. the degree of infection increases from one time step to the next by a
constant amount g ~ I%1.
Furthermore, another interaction between
neighbouring cells is introduced at this place. Infected neighbouring ,'ells can be synchronized by a
local averaging process. The state of an infected
cell at ti~:e : + 1. !~ then given by the sum of two
numbers: the mean degree of infection in the celt's
213
M. Gerhardt and H. Schuster / A cellular automaton
neighbourhood at time t and the constant g which
determines how quickly a cell passes through the
infected state.
To determine this mean degree of infection we
have to calculate the sum of the degrees of infection in the cell's neighbourhood S~j(t), given by
the formula
Sii(t) =
E
x,.(t),
In summary, the state evolution of a cell is
defined by
K,j(t)
for x,j(t) = 0 ,
%(t+])--
(9)
L/,j(t) ] +g'v }
(6)
(r, s) neighb. (i, j )
O<x,,(t)< V
for0 <
0
and to divide this expression by the number of
infected cells in its neighbourhood l,j(t), as defined in (4). The maximal degree of infection of a
cell is bounded by the value V, i.e. if the calculated degree of infection would pass the value V,
this value V is just taken as the new state of the
cell. All together, the state of an infected cell at
time t + 1 is given by
x,j(t+l)=min
for 0 <
{[s,,(')l
[ ~ 1
+g,V}
(7)
xu(t ) < V,
with I,j(t) and S,j(t) as defined in (4) and (6),
and g ~ N .
The local averaging process, introduced at this
place, can be chemically interpreted as heat transfer between neighbouring catalytic units which
develop into the palladiumoxide phase and are
thus forced to act synchronously.
3.3.
The evolu'ion of an ill cell
An ill cell can revert tc the healthy state in a
single step, i.e.
x,j(t + 1)
=0
for
x,)(t)=
l/.
(8)
This transition rule describes, in an abstract way,
the reduction of a catalytic unit after the paliadiumoxide phase is reached.
xij(t) < V,
forxi~(t) = V,
using the constants V, g, k t and k 2 E N and the
expressions K,j(t), l,j(t) and S,~(t) as defined in
(3), (4) and (6).
4. Results
The numerical results presented in the following
describe the dynamical behaviour of the cellular
automaton with four hundred cells placed on a
20 × 20 lattice with the Moore neighbourhood.
This two-dimensional field of cells is not closed to
a toms, i.e. the cells at the border of the field have
fewer neighbours than the ceils in its interior. The
constants II, k t and k 2 are fixed by V=100,
k 1 =- 2 and k 2 = 3, while the constant g is varied
between g = 1 and g = 20. Startirg with randomly
determined initial states the time evolution of the
cellular automaton is computed over 10000 time
steps.
In a first step of the numerical investigation the
time evolution of the coverage of the field with
infected cells is calculated. This output function is
chosen, because it enables a comparison between
the experimental measurements and the numerical
results of the cellular automaton to be made: An
infected cell represents a catalytic unit which develops into the palladiumoxide phase. Since the
oxidation of a palladium crystallite is correlated to
a higher rate of reactivity, the crystallites developing into the palladiumoxide phase are mainly responsible for the CO 2 production of the catalytic
214
M. Gerhardt and H. Schuster / A cellular automaton
type 2
type 1
1.0
0,-~
0,5
rj
•
400
500
. _
.
600
.
700
.
800
.
0,0
,
.
3100
t
3200
3300
type 3
3400
3500
t
type4
1.0
1,0"
"0
0.5
°--
8~
0.0--
1900
2000
2100
2200
2300
0.0
.
.
.
1O0
.
.
.
200
.
.
300
.
.
.
400
.
.
.
500
t
Fig. 3. The four different types of behaviour occurring in tlae time evolution of the cellular automaton depicted by the coverage of
the field w~th infected cells.
system. The coverage of the field with infected
cells can therefore be correlated with the empirically observed conversion of CO.
In a few cases-especially for small values of
the constant g - t h e infection process "dies out"
after a certain time, i.e. the cellular automaton
reaches a quiesc~mt state in which all cells remain
healthy. In all ot~aer cases the time evolution of the
cellular automaton can be classified into four qualitatively defined types each of which shows quite a
different bel',aviour. These four types can be characterized in the following way (see fig. 3):
Type 1: Small deviations from the maximal intensity of coverage of the field with infected cells occur at irregular intervals of
time. Larger deviations are occasionally
observed.
Type 2: Peaks of different amplitude occur at
nearly regular intervals of time. These
peaks show a typical saw-tooth pattern.
Type 3: Large peaks of practically the same amplitude occur at nearly regular intervals
of time.
Type 4: Small deviations with an average of
around 0.75 occur during the time evolution of the coverage of the field with
infected cells.
The development of these four types depends
upon the values of the different system parameters, mainly upon the value of the constant g. For
small values of g only the behaviour of type 1 or
type 2 can be observed, large values of g lead to a
behaviour of type 3 or 4 (cf. fig. 4). By fixing a
c.rtain value of g, ~t is possible to observe tranA-
M. Gerhardt and H. Schuster / A cellular automaton
I
• I I
5b I
4
?
n,.
•
j
I0
15
Fig. 4. The ranges of the control parameter g where the
various types in the behaviour of the cellular automaton have
been observed.
o
g
,,
|,*
/11
III
IN
INI
IIN
F i g 5. 'rime evolution of the cellular automaton in which the
system switches from type 3 to type 4, and back again.
tions between the different types. A time evolution
is shown in fig. 5, in which, for g - - 15, the system
switches over from type 3 to type 4 and back
again.
The behaviour of the cellular automaton can be
classified into four different types not only according to the time evolution of the coverage of the
field with infected cells, but also according to the
spatial patterns which evolve on the field. Each of
the four types is correlated to certain spatially
ordered structures, which can be characterized by
different forms of transition waves. While such a
transition wave is spreading over the field, in-
215
out from a central focus across the field. A typical
time evolution of such a transition wave is shown
in fig. 6. In this representation the state value of
each cell of the two-dimensional lattice is associated with the z-coordinate. At time step t = 1997 a
transition wave starts to spread out from the middle of the field. (The deep valley in the middle of
the field indicates the area where the transition
wave starts.) In the following time steps the transition wave travels over the field fight up to the
border. At time step t = 2003 all cells have passed
through the ill state, end only "newly" infected
cells can be found on tne field reverting to the ill
state again. At time step t = 2016 a new transition
wave starts. Qualitatively, the form of this ensuing
transition wave is similar to the previous one. This
regularity of the different transition waves is typical for type 3.
In type 2 the same form of circular transition
waves- starting from a central focus and moving
".:ut over the field up to the b o r d e r - c a n be observed. But as opposed to type 3, where the size of
the regions of cells passing through the ill state at
the same moment remains nearly constant from
one transitiop wave to the next, the size of these
regions varies in type 2. First, the areas of cells
passing through the ill state simultaneously expand from one transition wave to the next, until
the synchronized regions reach a certain size. The
size of the synchronized regions of the ensuing
transition wave is distinctly smaller. The process
of expansion of the synchronized areas is again
repeated.
The time evolution of the transition waves of
type 4 is different to those of types 2 and 3. In this
case, the transition waves travel over the field in
i ~ . t ~ t t ~.ezia l~a~.ll i.he m sIaT.c a l i ~ a,~. u,~.u ~.t,J,-
l, l l t $ . ,
vetted into the healthy state. (Since the number of
infected cells decreases during the propagation of
a transition wave, one can simultaneously observe
peaks of decrease in the time evolution of the
coverage of the field with ipfected cells.)
The particular form of the transition waves is
different for each of the four types. In type 3 one
can observe circular transition waves spreading
spira! transition wave is shown in fig. 7. Here the
core of the spiral wave is in the middle o r the ,,:cid.
In typc 1 the transition waves do not spread
across the whole field. They only spread over
small regions and then vanish. We call this third
kind of wave "meandering waves." At every time
step the state of the field appears son,~what inhomogeneous in comparison with the other types.
IK/EllX
K,tl
fall
d~l.dllC.ltl,
ltl~
~,ltzlltql~
~V~Jll.llr*l~'ll
g'l
, ~ s l
216
M. Gerhardt and H. Schuster / A cellular automaton
~
t
t
~
i ;;!.
""
: le13
:
20f6
iii' /
Fig. 6. A typical lime evolution of lhe infqction slates of the ct'll~ corresponding to lypc 3.
M. Gerhardt and H. Schuster/ A cellular automaton
217
Fig. 7. A typical time evolution of the infection states of the cells corresponding to ,'ype 4.
Transitions in the behaviour of the cellular aut o m a t o n from one type to the other are strongly
correlated to a corresponding change in the spatial
patterns which evolve on the field. As an example,
the observed transitions between type 3 and type
4, depicted in fig. 5, are caused by the change from
a circular :.o a spiral transition wave and back
again.
The formation of the spatial patterns, as described above, can also be observed with a different form of neighbourhood (e.g. von N e u m a n n
neighbourhood) or with a ditterent topological
structure of the field (e.g. a toroidal lattice).
The qualitative behaviour of the cellular automaton depends on the values of the constants k~
and k 2 which determine the intens;,ty of the infection process (see table I). For large values of k~
and k 2 the intensity of the infection process is so
weak, that for all randomly determined initial
states the infection dies out after a certain time.
While for smaller values of k I and k 2 (i.e. k 1 _< 4
and k 2 _< 3) the behaviour of the cellular automa-
ton can be described by one of ~he JO~,
e , , ~.ypcs,
large values of k 2 (k 2 > 4) and small values of k t
(k t < 2) lead to new phenomena. One can observe,
in this case, different isolated regions of infected
cells on the field during the time evolution. Within
these regions transition waves similar to those
des :fibed alcove occur.
Calculations were also made using a higher
number of cells, up to 250000 cells which were
placed on a 500 x 500 lattice. The spatial patterns
can be characterized, in all these cases, by the
aforementioned form of circular, spiral and meandering transition waves, even if the number of loci
from which the transition waves spread out usu,~lh, ;+,_,',*.c,,~e,~e ~ a , ; t h
~n
i n n ~ ' , ~ e l n a nHrr~hor c~f co1|~
An impression of the fascinating variety of spatial patterns which are produced by the cellul,%,automaton is given by some typical infection states
of the field, depicted in plate I. In this re~resentatior the state value of each cell is associated with, a
certain color. The results, shown in this fignre, are
obtained with an extended form of the Moore
218
M. Gerhardt and H. Schuster/ A cellular automaton
Table I
Influence of the values of the r, astants k~ and k 2 on the dynamical behavioar of the cellular automation.
k2•
~:1 1
1
2
3
4
>_5
type 2
type 3
type 2
typc?
type 2
cype 3
type 2
type 3
ce~afion
type 2
type 3
type 2
type 3
lVpe 2
I VI~ 3
type 2
type 3
cessation
of the
o f t he
infection
proe¢~
infection
process
>4
type I
type 2
type 3
type 4
type 1
type 2
type 3
type 4
type 1
type 2
type 3
type 4
type I
type 2
type 3
type 4
cessation
of the
infection
process
isolated
regions
of infection
isolated
regions
of infection
cessation
of the
infection
process
cessation
of the
infection
process
cessation
of the
infection
process
neighbourhood in which the neighbourhood includes not only the eight nearest neighbouring
cells, but, in addition, also the sixteen next to
nearest neighbours.
5. Discussion
In the previous section, it has been demonstrated that the time evolution of the cellular
automaton leads to highly ordered spatio-temporal patterns. Its qualitative behaviour can be
characterized by only four different types. Each of
these types corresponds to the formation of certain spatial structures, such as circular or spiral
waves propagating across the field.
As desctSbed above, one can discuss the agreement between these numerical results and the experimental obse~ations of the CO orddation- the
starting-point of our work - by comparing the evolution of the coverage of the field with infected
cells with the empirically observed conversion of
CO. In view of the very abstract nature of the
mathematical description one cannot expect an
exact quantitative agreement. But, in at least a
qualitative sense the cellular automaton exhibits
almost the same characteristics in its time evolution as the catalytic system. Depending on the
different experimental parameters the chemical
system produces a variety of oscillation patterns,
like homogeneous oscillations, fractal-like oscillations (cf. fig. 1) or quasi stationa.. 3, behaviour with
only small fluctuations. All these phenomena are
also reproduced by the cellular automaton in the
four different types of behaviour. Moreover, the
agreement between the theoretical and the experimental system is confirmed by the fact that a
variation of related parameters in both systems
leads to the same qualitative beha,dour. (A detailed discussion of the a~eement between the
experimental system and the cellular automaton is
given in [31].)
Similar oscillation lZatterns are also observed in
other catalytic oxidation reactions. As already desc~bed in section 2 different reaction mechanisms
are d;,scussed as an explanation for the oscillatory
M. Gerhardl and H. Schuster/ A cellular automaton
219
Plate I. Some typical infection states in the time evolution o f the cellular automaton: (a) circular waves on a 100x 100 lattice and (b~
on a 250X25G latlice~ (c) inhomogeneous slate oftype I on a 2 5 0 X 2 5 0 lattice; (d,e) spiral waves on a 250:<250 lattice and (f) on a
500 X 500 lattice. The coiours associated with the different states o f infected cells vary from blue to red in a, b, c and f. and f~om dack
bhle lo light blue in d and e. Furthermore, in b each state vahqe of an infected cell which is divisible by 4 is coloured black.
220
M. Gerhardt and H. Schuster/A cellular automaton
behaviour of these catalytic systems. But nearly all
these reaction schemes agree in the assumption of
a cyclic transition between different reaction
branches. Even while the cellular automaton is
derived from a specific ~,xperimental situation, the
rules of the evolution of a single cell are just based
on this idea of a transition between different states
of activity and might, therefore, also be relevant
for other catalytic systems.
The observed variety of phenomena of the cellular automaton is not only the result of the typical
transition cycle in the evolution of a single cell,
but, above all, the outcome of an interaction process between a large number of cells. Together
with the experimental observations of spatial patterns on the catalyst surface in heterogeneous catalytic reactions [4, 6] and related theoretical models [7, 28] this cellular automaton can confirm, as
we think, the chemical concept that oscillation
patterns observed in diff~Tent catalytic systems
might be correlated to the formation of spatially
ordered structures on the catalyst, which are
caused by local interactions between independen,ly acting catalytic units.
While some of the patterns which occur in the
time evolution of the cellular automaton are similax to those observed by Schmitz et al. in heterogeneous catalytic systems [4], others show a
surprising similarity to the experimental obse~,ations of the Belousov-Zhabotinskii reaction, especially wi~ regard to the formation of spiral waves
(cf. [16-19]).
The phenomena of pattern formation in excitable media, as in the Belousov-Zhabotinskii
reaction is not only often investigated experimentally, but also a lot of theoretical work was done
in this field. In this context, the chemical waves
are usually described by partial differential equations, e.g. in form of reaction--diffusion equations.
Even if these nonlh~ear equations can seldom be
solved in closed form, a variety of approximation
and numerical techniques could successfully be
used to. show that spiral and circular waves represent typical ph~.tmmena in these chemical systems.
Reviews on the theory of reaction-diffusion equa-
tions and its applications can be found in [32-36].
Besides this approach of partial differential equations other authors have already suggested models
-in the form of cellular automata as a theoretical
description of pattern formation in excitable media [37-39].
There is, of course, a strong correspondence
between partial differential equations and cellular
automata, because the result of each discretization
of a partial differential equation is a cellular automaton. Since the cellular automaton, introduced
in this article, shows the typical phenomena of
circular and spiral waves in a surprising similarity
to those observed in the Belousov-Zhabotinskii
reaction, the question arises whether there might
be a connection between the specific rules of the
"hodge podge machine" and the well-known description of the form of reaction-diffusion equations. Although we cannot as yet answer this
que,~tion, we believe that it might be interesting
for future discussion. On one side one might be
able to learn something new about these phenomena, looking at them from a different point of
view. On the other hand, the "hodge podge machine" can be implemented so easily on conventional digital computers that it might be an
appropriate approximation tool for numerical simulations of such natural systems.
Besides the relevance of this cel!uJar automaton
to chemical systems, we believe that this model is
an interesting example of cellular automata which
lead to a self-sustained organization of a small
number of characteristic spatio-temporal patterns.
It should be stressed once more that the formation
of these patterns is always the outcome of time
evolutions which start from random(¥ determined
initial states. In other words, the input of the
"hodge podge machine" is a completely disordered state, and after the "machine" has worked
for some time, highly ordered states occur.
Even if we do not know up to now any mathematical ~heory which could be used to prove rigorously that the rules defining the "hodge podge
machine" will always lead to such patterns, these
patterns are so fascinating in themselves, that we
M. Gerhardt and H. Schuster/ A cellular automaton
believe the "hodge podge machine" might be an
interesting example for those who are interested in
the getaeral field of cellular automata.
Acknowledgements
We wish to thank Andreas Dress, as well as Nils
Jaeger, and Peter Plath for many stimulating discussions during the course of our work. The
eoloured computer graphics of plate I have been
worked out during a research visit of one of the
authors at the Max-Planck-lnstitut f'fr Erniihrungsphysiologie in Dortmund, which was supported by a grant of the Stiftung Volkswagenwerk
and for which we want to thank the director of the
institute, B. Hess, and the Stiftung Volkswagenwerk. We are also grateful to the members of the
institute, in particular to M. Markus and S.C.
Miiller, for helpful comments. Financ~.al support
from the Westf'alisch-Lippische Universit~itsgesellschaft for the costs of printing the coloured graphics of plate I is gratefully acknowledged. For reading and correcting this paper we are indebted to
G.S. Macpherson.
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